Volterra-Choquet nonlinear operators
DOI:
https://doi.org/10.12775/TMNA.2020.009Keywords
Choquet integral, monotone, submodular and continuous from below set function, Choquet $L^{p}$-space, distorted Lebesgue measures, Volterra-Choquet nonlinear operator, compactness, Lipschitz properties, cyclicityAbstract
In this paper we study to what extent properties of the classical linear Volterra operators can be transferred to the nonlinear Volterra-Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclicity properties are studied.References
D.R. Adams, Choquet integrals in potential theory, Publ. Mat. 42 (1998), 3–66.
O. Baksi, T. Khan, J. Lang and V. Musil, Strict S-numbers of the Volterra operator, Proc. Amer. Math. Soc. 146 (2018), no. 2, 723–731.
M.S. Brodskiı̆, On a problem of I.M. Gelfand, Uspekhi Mat. Nak. 12 (1957), 129–132.
M. S. Brodskiı̆, Triangular and Jordan Representation of Linear Operators (translated from Russian by J.M. Danskin), Translations of Mathematical Monographs, vol. 32, American Mathematical Society, Providence, R.I., 1971.
J. Cerdà, J. Martı́n and P. Silvestre, Capacitary function spaces, Collect. Math. 62 (2011), 95–118.
G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1954), 131–295.
G. Choquet, La naissance de la théorie des capacités: réflexion sur une expérience personnelle, Comptes rendus de l’Académie des sciences, Série générale, La Vie des sciences, 3 (1986), 385–397.
J.B. Conway, A Course in Operator Theory, Rhode Island, Amer. Math. Soc., 2000.
D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht, 1994.
J. Dixmier, Les operateurs permutables à l’operateur integral, Portugal Math. 8 (1949), 73–84.
W.F. Donoghue, The latice of invariant subspaces of quasi-nilpotent completely continuous transformation, Pacific J. Math. 7 (1957), 1031–1035.
N. Dunford and J.T. Schwartz, Linear Operators, vol. 1, Wiley–Interscience, 1958.
A.F.M. ter Elst and J. Zemánek, Contractive polynomials of the Volterra operator, Studia Math. 240 (2018), no. 3, 201–211.
S.G. Gal, Approximation by Choquet integral operators, Ann. Mat. Pura Appl. 195 (2016), no. 3, 881–896.
S.G. Gal, Uniform and pointwise quantitative approximation by Kantorovich–Choquet type integral operators with respect to monotone and submodular set functions, Mediterr. J. Math., 14 (2017), no. 5, art. 205, 12 pp.
S.G. Gal, Quantitative approximation by nonlinear Picard–Choquet, Gauss–Weierstrass–Choquet and Poisson–Cauchy–Choquet singular integrals, Results Math. 73 (2018), no. 3, art. 92, 23 pp.
S.G. Gal, Fredholm–Choquet integral equations, J. Integral Equations Appl. 31 (2019), no. 2, 183–194.
S.G. Gal, Volterra–Choquet integral equations, J. Integral Equations Appl. 31 (2019), no. 4, 495–504.
S.G. Gal and B.D. Opris, Uniform and pointwise convergence of Bernstein–Durrmeyer operators with respect to monotone and submodular set functions, J. Math. Anal. Appl. 424 (2015), 1374–1379.
S.G. Gal and S. Trifa, Quantitative estimates in Lp -approximation by Bernstein–Durrmeyer–Choquet operators with respect to distorted Borel measures, Results Math. 72 (2017), no. 3, 1405–1415.
E. Gallardo and A. Montes, Volterra operator is not supercyclic, Integral Equations Operator Theory 50 (2004), no. 2, 211–216.
I.C. Gohberg and M.G. Krein, Theory and Applications of Volterra Operators in Hilbert Spaces (translated from Russian by A. Feinstein), Translations of Mathematical Monographs, vol. 24, American Mathematical Society, Providence, R.I., 1970.
P.R. Halmos, A Hilbert Space Problem Book, New York, Springer Verlag, 1967.
K.G. Ivanov, On a new characteristic of functions, II. Direct and converse theorems for the best algebraic approximation in C[−1, 1] and Lp [−1, 1], Pliska Stud. Math. Bulgar. 5 (1983), 151–163.
G.K. Kalisch, On similarity, reducing manifolds, and unitary equivalence of certain Volterra operators, Ann. Math. 66 (1957), 481–494.
L. Khadkhuu and D. Tsedenbayar, A note about Volterra operator, Math. Slovaca 68 (2018), no. 5, 1117–1120.
L. Khadkhuu and D. Tsedenbayar, On the numerical range and numerical radius of the Volterra operator, The Bulletin of Irkutsk State University, Series Mathematics 24 (2018), 102–108.
P. Lefèvre, The Volterra operator is finitely strictly singular from L1 to L∞ , J. Approx. Theory 214 (2017), 1–8.
F. León-Saavedra and A. Piqueras-Lerena, Cyclic properties of Volterra operator, Pacific J. Math. 211 (2003), no. 1, 157–162.
F. León-Saavedra and A. Piqueras-Lerena, Super convex-cyclicity and the Volterra operator, Monatsh. Math. 177 (2015), no. 2, 301–305.
Yu-Xia Liang and Rongwei Yang, Energy functional of the Volterra operator, Banach J. Math. Anal. 13 (2019), no. 2, 255–274.
G. Little and J.B. Reade, Estimates for the norm of the n-th indefinite integral, Bull. London Math. Soc. 30 (1998), 539–542.
N.H. Salas, Supercyclicity and weighted shifts, Studia Math. 135 (1999), 55–74.
D. Sarason, A remark on the Volterra operator, J. Math. Anal. Appl. 12 (1965), 244–246.
Gh. Siretchi, Differential and Integral Calculus, vol. II, Edit. Stiint. Enciclop., Bucharest, 1985.
M. Sugeno, A note on derivatives of functions with respect to fuzzy measures, Fuzzy Sets and Systems 222 (2013), 1–17.
D. Tsedenbayar, On the power boundedness of certain Volterra operator pencils, Studia Math. 156 (2003), 59–66.
R.S. Wang, Some inequalities and convergence theorems for Choquet integrals, J. Appl. Math. Comput. 35 (2011), 305–321.
Z. Wang and G.J. Klir, Generalized Measure Theory, Springer, New York, 2009.
Z. Wang and J.-A. Yan, A selective overview of applications of Choquet integrals, Adv. Lect. Math., Springer, 2007, pp. 484–515.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
